Estimate Area Using Rectangles and Right endpoints

Question

Estimate the area under the graph of the function f(x) = 1/x on the interval [1,2] using 3 equal subintervals and right endpoints evaluation.

How do I do this? I got subintervals of (1,1),(1.5,1/1.5),(2,1/2)

Now what? Or is that wrong?

Answer 1

You ought to find the red area.

It's a collection of three rectangles having following dimensions:

1. $f(1+1/3) \times 1/3 = $ height $\times$ width of tallest rectangle

2. $f(1+2/3) \times 1/3 = $ height $\times$ width of central rectangle

3. $f(1+3/3) \times 1/3 = $ height $\times$ width of shortest rectangle

Answer 2

Width of the interval $[1,2]$ is 1. So, shouldn't your sub-intervals be $[1 , 1+1/3]$, $[1+1/3 , 1+2/3]$, & $[1+2/3 , 2]$?

Answer 3

The area is approximated as

$$A = \Delta x \left[f(x_1) + f(x_2) + f(x_3 )\right]$$

$$ = \frac13 \left[f(\frac43) + f(\frac53) + f(2) \right] =\frac13 \left(\frac34 + \frac35 + 2 \right)=0.617$$

compared with the exact integration result $\ln2 = 0.693$.